'296 CtlLESTIAL MECHANICS. I. Vli. 34. 



becomes— ^^V + n^ (y^i/-hz^z); and the quan- 

 tity u^:v+v^y^w^z is not an exact variation. 



Supposing X to be the axis of motion, and the angular 

 velocity n, at a distance considered as unity, we shall 

 have V— — 7iz, wzzny, and the equation (JJ)(366) becomes 



^P t>T^ d'v ^ d'w rv rsxr c> 



p Qz '^ ay ^ ^7 



n,nz.^z'=.^V-\-n^{ij'^y-{-z^z): an equation of which both 



the members are exact variations, and which is therefore 

 possible. The equation (K) (367), will become 0=:Af 



~~+ uM -r^ + vAt--^ +wAt-,r^: and it is obvious that this 

 d^ do: dy dz 



equation will be satisfied if the fluid is homogeneous. Both 

 equations therefore being true, the supposed motion is 

 possible, and a fluid may move uniformly round an axis, 

 [without any internal change of the disposition of its par- 

 ticles.] 



The centrifugal force, at the distance >^(y^ + z*)from the 

 axis of rotation, is expressed by the square of the velo- 

 city n\y"-\-z^)y divided by the distance; consequently the 

 quantity n^{y^y+z^z) is the product of the centrifugal 



force ri^s/Cy^+z^) into the element of its direction • ■ ^ „ 



it is evident, therefore, by comparing this equation with 

 the general equation of the equilibrium of fluids in § 17 

 (316) that the conditions of the motion are reduced to 

 those of the equilibrium of a fluid actuated by the same 

 forces, and by the centrifugal force in addition to them : 

 which is also sufficiently obvious from the nature of the 

 case. 



